Form a circular Mesh out of an object
Some tips concerning the circular cut, starting from the answer indicated by Duarte Farrajota Ramos in the comments.

The result is the following, and this answer is concerning the cut only and its cleanup, not the logo part (as the approach may be very depending on the logo shape itself):



Just to say, we can notice that the starting mesh is very clean and uses a subdivision modifier so that we have a reasonable amount of vertices to work on:



After the cut is done, we obtain some distortions, as the cutting circle geometry does not correspond to the bottle. This is particularly visible without the subsurface modifier:



So, the cleanup consists mainly on three actions (and this works because we know that we'll have the subdivision modifier after that):


Dissolving unwanted vertices
Merging vertices
Adjust cut position


Dissolving vertices:


Select them
Use X and choose 'dissolve vertices'
The aim is to do that 'everywhere' to make quads again around the hole (again, we know that subdivision surface modifier will compensate that)




Mergin vertices:

At some places this is easier to merge AltM, either at center or at the last selected.



Adjust cut position:

Select the wanted vertex and use G twice to move it along the edge:



Once all done, in this case, remove the inner part of the cut:



Now select the border, extrude E, and scale S a bit. The aim for that is to isolate the central part with a 'clean' edge loop:



Last steps, filling the hole:



Continue that as possible and finally adjust vertices in the same way as the beginning so that almost all faces are quads:



Link to the blend file